How do I constrain location to a path
Not sure if this answers your question but if you add a follow path constraint to the empty, check fixed position and then move the offset from zero to one the empty is locked on to the curve which now functions as it's path. I really hope this helps and I didn't misunderstand the question.

(I then, just for effect, parrented the cylinder to the empty, to visualize a handle.)

Here's a picture of the setup.